Trivial Ring Extensions Defined by Arithmetical-Like Properties

In this article we investigate the transfer of the notions of elementary divisor ring, Hermite ring, Bezout ring, and arithmetical ring to trivial ring extensions of commutative rings by modules. Namely, we prove that the trivial ring extension R: = A ⋉ B defined by extension of integral domains is an elementary divisor ring if and only if A is an elementary divisor ring and B = qf(A); and R is an Hermite ring if and only if R is a Bezout ring if and only if A is a Bezout domain and qf(A) = B. We provide necessary and sufficient conditions for R = A ⋉ E to be an arithmetical ring when E is a nontorsion or a finitely generated A − module. As an immediate consequences, we show that A ⋉ A is an arithmetical ring if and only if A is a von Neumann regular ring, and A ⋉ Q(A) is an arithmetical ring if and only if A is a semihereditary ring.